/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 Sketch each angle in standard po... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 21

Sketch each angle in standard position. (a) \(-30^{\circ}\) (b) \(150^{\circ}\)

Short Answer

Expert verified
The sketch for -30 degrees will be a line from the origin at an angle of 30 degrees to the positive x-axis in the clockwise direction. The sketch for 150 degrees will be a line from the origin at an angle of 150 degrees to the positive x-axis in the counter-clockwise direction.

Step by step solution

01

Sketch for -30 degrees

First, start with the initial side on the positive x-axis, which is the standard position. Since -30 degrees is a negative angle, it will be measured in a clockwise direction from the initial side. Draw a line from the origin forming an angle of 30 degrees with the positive x-axis in the clockwise direction.
02

Label the -30 degrees sketch

After drawing the line, annotate the sketch to indicate that the angle is -30 degrees. At this point, you should have a sketch of an angle at -30 degrees in the standard position.
03

Sketch for 150 degrees

Again, start with the initial side on the positive x-axis, which is the standard position. Since 150 degrees is a positive angle, it will be measured in a counter-clockwise direction from the initial side. Draw a line from the origin forming an angle of 150 degrees with the positive x-axis in the counter-clockwise direction.
04

Label the 150 degrees sketch

After drawing the line, annotate the sketch to indicate that the angle is 150 degrees. At this point, you should have a sketch of an angle at 150 degrees in the standard position.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Position
In geometry, when we talk about angles, it's important to place them in a way that's easy to understand.
This brings us to the concept of the "Standard Position". Standard position is a way to make sure we're all on the same page when sketching or measuring angles.
Here's what you need to know about it:
  • The angle's vertex is at the origin of a coordinate plane.
  • One side of the angle (the initial side) rests along the positive x-axis.
  • The other side (the terminal side) is what's measured to determine the angle's size.
By using standard position, anyone looking at a diagram will know exactly where the angle starts and where it ends.
It's like having a universal starting point, making communication in math much clearer!
Positive Angle
Positive angles are a fundamental concept in understanding angle measurement.
Here's how they work:
  • They are measured starting from the initial side on the positive x-axis.
  • The measurement is made in a counterclockwise direction.
  • Common positive angles include familiar ones like 90掳, 180掳, and 360掳.
This counterclockwise measurement not only helps in defining what a positive angle is, but it also sets the scene for distinguishing between positive and negative angles.
In sketches, a positive angle will look like it's opening upwards from the x-axis toward the top of the page.
Negative Angle
When it comes to negative angles, the clock is essentially turned backwards.
Negative angles are measured differently from positive angles, and here's how:
  • They start from the positive x-axis just like positive angles.
  • The twist is that they're measured in a clockwise direction.
  • Negative angles may not be as intuitive but are just as important in geometry, helping describe motion or orientation in reverse.
For example, a -30掳 angle begins at the positive x-axis and turns downwards.
Negative angles are just another way angles can be utilized to describe a full 360掳 rotation in any direction.
Clockwise and Counterclockwise Measurement
Clockwise and counterclockwise directions are key in understanding how angles are measured.
Let's break it down:
  • Clockwise direction is akin to the movement of a clock's hands, sweeping from the top towards the right, down, and around.
  • In geometry, an angle measured in this direction is negative.
  • On the flip side, counterclockwise direction moves opposite to a clock's hands.
  • An angle measured in this manner is positive.
Understanding these directions helps you visualize and sketch angles accurately.
It's a simple yet important concept, ensuring clarity whether you're drawing an angle or indicating a rotation.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by \(y=\frac{1}{4} \cos 16 t, \quad t>0.\) where \(y\) is measured in feet and \(t\) is the time in seconds. (a) Use a graphing utility to graph the function. (b) What is the period of the oscillations? (c) Determine the first time the ball passes the point of equilibrium \((y=0).\)

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. arccos( -0.9 )

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$395^{\circ}$$

The front of an A-frame cottage has the shape of an isosceles triangle. It stands 28 feet high and the angle of elevation of its roof is \(70^{\circ} .\) What is the width of the cottage at its base?

The table shows the average daily high temperatures (in degrees Fahrenheit) for Quillayute, Washington, \(Q\) and Chicago, Illinois, \(C\) for month \(t,\) with \(t=1\) corresponding to January. $$\begin{array}{c|c|c} \text { Month, } & \text { Quillayute, } & \text { Chicago, } \\ t & Q & C \\ \hline 1 & 47.1 & 31.0 \\ 2 & 49.1 & 35.3 \\ 3 & 51.4 & 46.6 \\ 4 & 54.8 & 59.0 \\ 5 & 59.5 & 70.0 \\ 6 & 63.1 & 79.7 \\ 7 & 67.4 & 84.1 \\ 8 & 68.6 & 81.9 \\ 9 & 66.2 & 74.8 \\ 10 & 58.2 & 62.3 \\ 11 & 50.3 & 48.2 \\ 12 & 46.0 & 34.8 \end{array}$$ (a) \(\mathrm{A}\) model for the temperature in Quillayute is given by $$Q(t)=57.5+10.6 \sin (0.566 x-2.568)$$ Find a trigonometric model for Chicago. (b) Use a graphing utility to graph the data and the model for the temperatures in Quillayute in the same viewing window. How well does the model fit the data? (c) Use the graphing utility to graph the data and the model for the temperatures in Chicago in the same viewing window. How well does the model fit the data? (d) Use the models to estimate the average daily high temperature in each city. Which term of the models did you use? Explain. (e) What is the period of each model? Are the periods what you expected? Explain. (f) Which city has the greater variability in temperature throughout the year? Which factor of the models determines this variability? Explain.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.